Department: As far as the convention structure, your department is Programming. The form is asking which part of Programming you think your event fits best into. This is kind of a carry over from the past since almost everyone's events nowadays are panels. Here's some things you might want to note if you have an idea other than a panel. If you want to run a tournament for a game that would be under either Video Gaming or Tabletop Gaming. If you want to run a Demonstration or How-To panel, that might be under the Creation Station. If you want to put run a costume contest of some kind that has to do with cosplay.
Day/Time: If you want it before 3pm than I suggest writing "Sunday Before 3pm, Saturday before 3pm, or Monday before closing" and then just choose what order to put the days in. I suggest Sunday as your top choice.
Room Style: Theater style is what you want.
Membership: That I cannot help you with, but I was told that you need to have 4hours of content at the con in order to qualify for panelist discount. There's a number but I don't have it. That's something we'd need to get from a Programming manager. I'm just a staffer with lots of panel-running experience.
Side note, if you are a panelist with lots of stuff you need to get to your panel room. I suggest writing in that you need a volunteer to help with set up, that way they can schedule someone to help you schlep on down to the panel rooms. They are in a high traffic area this year.